Thermophoresis beyond Local Thermodynamic Equilibrium

We measure the thermophoresis of polysterene beads over a wide range of temperature gradients and find a pronounced nonlinear phoretic characteristic. The transition to the nonlinear behavior is marked by a drastic slowing down of thermophoretic motion and is characterized by a Péclet number of order unity as corroborated for different particle sizes and salt concentrations. The data follow a single master curve covering the entire nonlinear regime for all system parameters upon proper rescaling of the temperature gradients with the Péclet number. For low thermal gradients, the thermal drift velocity follows a theoretical linear model relying on the local-equilibrium assumption, while linear theoretical approaches based on hydrodynamic stresses, ignoring fluctuations, predict significantly slower thermophoretic motion for steeper thermal gradients. Our findings suggest that thermophoresis is fluctuation dominated for small gradients and crosses over to a drift-dominated regime for larger Péclet numbers in striking contrast to electrophoresis.

Figure 1

Trajectories obtained from single-particle tracking of polystyrene beads (PSB) immersed in water for two different Péclet numbers (a) $\mathrm{Pe}=0.5$ and (b) $\mathrm{Pe}=5.0$. The scale bar corresponds to $20\mathrm{\mu }\mathrm{m}$. For $\mathrm{Pe}\lesssim 1$, the motion is dominated by diffusion and thermophoresis is rather slow. In contrast, for $\mathrm{Pe}\gtrsim 1$ hydrodynamic stresses become important, increasing the phoretic drift velocity.

Figure 2

Linear fits to the experimental measurements (symbols) for radii (a) $a=550$ and (b) $a=950\mathrm{nm}$ and for different salt concentrations $c$. The crossover temperature gradients $|\nabla T{|}_c^{\exp }$ where the calculated velocities deviate by 70% from the linear drift relation are indicated by the vertical dashed lines. (c) Soret coefficient $S_T$ vs Debye length ${\kappa }^{-1}$. Full lines correspond to the theoretical predictions [see Eq. (4) in the Supplemental Material [35] ]. (d),(e) Comparison to the different linear-theoretical-model predictions for the same radii and concentrations. Vertical dashed lines represent the crossover temperature gradient $|\nabla T{|}_{\mathrm{c}}$ obtained from theory (see Table S1 in Supplemental Material [35]). Full lines denote results from both the thermodynamic and hydrodynamic approach by superimposing the corresponding thermophoretic drift velocities. Dashed-dotted lines are solutions within a hydrodynamic viewpoint. (f) Scatter plot of the crossover temperature gradient $|\nabla T{|}_{\mathrm{c}}^{\mathrm{fit}}$ obtained from linearly fitting the data according to Eq. (2) vs $|\nabla T{|}_{\mathrm{c}}$. Inset: Same for the experimentally determined thermal gradient $|\nabla T{|}_{\mathrm{c}}^{\exp }$ (see Table S1 in the Supplemental Material [35]).

Figure 3

Single master curve with pronounced nonlinear deviations setting in around $\mathrm{Pe}\gtrsim 1$ for all particle radii and salt concentrations (see also Table S1 in Supplemental Material [35]).